The equilibrium concentration of polyvacancy in crystal was formulated by modifying the grand canonical ensemble. The generalized equation could be applied to obtain the concentration of vacancy in any order and shape in crystal. In this work, a Widom-like particle insertion method was adopted implemented with molecular dynamics simulation to obtain individual formation free energies for constituent vacancies in polyvacancy. As a case study, equilibrium concentrations and formation free energies were obtained of four and nine trivacancies identified in face-centred cubic and hexagonal close-packed hard-sphere crystals, respectively. The result was in excellent agreement with literature data available for the face-centered cubic. Further, stabilities of equilateral triangular trivacancies were elucidated and their relative stabilities in face-centered cubic and hexagonal close-packed hard-sphere crystals were reported. Overall trivacancy concentration was higher in the hexagonal close-packed hard-sphere crystal.

Generalized Equilibrium Concentration of Polyvacancy: Case Study for Trivacancy in Hard-Sphere Crystals. C.Liu, P.C.Su, S.K.Kwak: Molecular Physics, 2011, 109[20], 2461-70